Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-26T08:36:52.288Z Has data issue: false hasContentIssue false
  • English
  • Français

The two-barrier problem for continuously differentiable processes

Part of: Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Igor Rychlik
Igor Rychlik*
Affiliation:
University of Lund
*
∗Postal address: Department of Mathematical Statistics, University of Lund, Box 118, S-221–00 Lund, Sweden.
Get access Rights & Permissions [Opens in a new window]

Abstract

An efficient algorithm to compute upper and lower bounds for the first-passage time in the presence of a second absorbing barrier by means of a continuously differentiable decomposable process, e.g. a smooth function of a continuously differentiable Gaussian vector field, is given. The method is used to obtain accurate approximations for the joint density of the zero-crossing wavelength and amplitude and the distribution of the rainflow cycle amplitude. Numerical examples illustrating the results are also given.

Keywords

CROSSINGSFATIGUEFIRST-PASSAGE DENSITYRAINFLOW CYCLEWAVEHEIGHTWAVELENGTH

MSC classification

Primary: 60G70: Extreme value theory; extremal processes
Secondary: 60G17: Sample path properties
Type
Research Article
Information
Advances in Applied Probability , Volume 24 , Issue 1 , March 1992 , pp. 71 - 94
Copyright
Copyright © Applied Probability Trust 1992 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Research supported in part by the Swedish Natural Science Research Council, Grant No F-PD 9280–100, and by the Air Force Office of Scientific Research, Grant No. F49620 85C 0144 at the University of North Carolina, Chapel Hill.

References

[1] Abrahams, J. (1986) A survey of recent progress on level crossing problems for random processes. In Communications and Networks. A Survey of Recent Advances , ed. Blake, I. F. and Poor, H. V., pp. 6–25. Springer-Verlag, New York.Google Scholar
[2] Blake, I. F. and Lindsey, W. C. (1973) Level-crossing problems for random processes. IEEE Trans. Inf. Theory 19, 295–315.Google Scholar
[3] Collins, J. A. (1981) Failure of Materials in Mechanical Design. Wiley-Interscience, New York.Google Scholar
[4] Cramer, H. and Leadbetter, M. R. (1967) Stationary and Related Stochastic Processes. Wiley, New York.Google Scholar
[5] Ditlevsen, O. (1985) Fatigue model for elastic bars in turbulent wind. J. Wind Engng. Industrial Aerodynamics 18, 27–52.Google Scholar
[6] Dowling, N. E. (1972) Fatigue prediction for complicated stress-strain histories. J. Materials 7, 71–87.Google Scholar
[7] Durbin, J. (1985) The first-passage density of a continuous Gaussian process to a general boundary. J. Appl. Prob. 22, 99–122.Google Scholar
[8] Ford, D. (1988) Range-Mean-Pair exceedances in stationary Gaussian processes. In Reliability and Optimization of Structural Systems, Aalborg, Denmark , pp. 119–139. Springer-Verlag, Berlin.Google Scholar
[9] Leadbetter, M. R., Lindgren, G. and Rootzén, H. (1983) Extremes and Related Properties of Random Sequences and Processes. Springer-Verlag, New York.Google Scholar
[10] Lindgren, G. (1983) Use and structure of Slepian model processes for prediction and detection in crossing and extreme value theory. Proc. NATO ASI on Statistical Extremes and Applications, Vimeiro, 1983 , pp. 261–284. Reidel, Dordrecht.Google Scholar
[11] Lindgren, G. and Rychlik, I. (1991) Slepian models and regression approximations in crossing and extreme value theory. Internat. Statist. Rev. 159, 195–225.Google Scholar
[12] Longuet-Higgins, M. S. (1983) On the joint distribution of wave periods and amplitudes in a random wave field. Proc. R. Soc. Lond. A 389, 241–258.Google Scholar
[13] Nielsen, R. J. (1988) Rainflow cycles in nonstationary Gaussian processes. Proc. Space '88 , Albuquerque, New Mexico.Google Scholar
[14] Rice, S. O. (1945) Mathematical analysis of random noise. Bell Syst. Tech. J. 24, 46–156.Google Scholar
[15] Rychlik, I. (1987) A note on Durbin's formula for the first passage density. Statist. Prob. Lett. 5, 425–428.Google Scholar
[16] Rychlik, I. (1987) Regression approximations of wavelength and amplitude distributions. Adv. Appl. Prob. 19, 396–430.Google Scholar
[17] Rychlik, I. (1988) Rain flow cycle distribution for ergodic load processes. SIAM J. Appl. Math. 48, 662–679.Google Scholar
[18] Rychlik, I. (1990) New bounds for the first passage, wave-length and amplitude densities. Stoch. Proc. Appl. 34, 313–339.Google Scholar
[19] Rychlik, I. (1990) Confidence bands for linear regressions. Univ. Lund Stat. Res. Rep. No. 1990:4, 1–15. Commun. Statist. To appear.Google Scholar